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The counterargument to Greg: the expected probability of finding the dot on the first page is your expected probability for the whole book divided by the number of pages. When you do not find a dot on the first page, you know nothing more about the book as a whole, so the probability on the second page is slightly higher -- book probability divided by pages minus one. It is an asymptotic function with a big divide-by-zero when you check the final page.
The flaw in Greg's argument is that the expected probability is not the same as the actual probability. Whether or not the dot is included was decided by principles well beyond your knowledge. IF reading a page made you reevaluate the content of the book, then I would agree with part of Greg's analysis. But Bayes' Theorem requires an input of how confident you are in the original estimate and how confident you are that new information should make you reevaluate confidence
If you check a whole book, then you would adjust up or down your confidence up or down for the next whole book from that publisher. But having checked a single page does not make you reevaluate *this* book.
Greg is correct.
Let's say you read 90% of the pages. You're saying there is a 90% chance the red dot is on one of the remaining pages. And, when you read all but the last page, there is a 90% chance it will be on the last page.
The logic flaw is that you do, in fact, learn something about the location of the red dot when you read half the book and don't find it. It's not like the Monty Hall problem, where you really don't learn anything about your choice when the goat is revealed.
Greg is correct that you should assume a 45% chance the dot is in each half of the book. When you find no dot in the first half of the book, there is a .45/.55 probability in the last half of the book.
No. I am NOT saying there is a 90% chance of the red dot being there. I am saying that *I EXPECT* there is a 90% chance of the red dot being there.
Those are two very different thing. I have no expectations when I start about which page the red dot will be on. None. So discovering that the dot is NOT on page 1 does not change my expectation about finding the dot *somewhere*.
Essentially, I am saying that I don't have a 90% expectation for any page *except* the last page that I check.
I think my chief disagreement here is When you do not find a dot on the first page, you know nothing more about the book as a whole ...but we do! We know it's not in the first 1/n of the book, which is more information. I'll agree that problems like this are very sensitive to how you frame them. But in this case, if we assume the red-dot-putter rolled a ten-sided die, and if it didn't come up 0 picked a random page to put the dot on, then Bayes' theorem applies. We aren't really dealing with how trustworthy the original 90% estimate is - that seems outside of the scope of the problem...
Your theorem leads to an incorrect belief in the following scenario:
Let's say the author writes 100 books. He chooses 90 of those books to put a red dot in. He does NOT roll a die. Instead, he puts the red dot on the last page of every single book.
If that's the case, then when you read half of every book, you would encounter the red dot on none of them. Under your rule, you would then assume that none of the books included a red dot, with near certainty.
You are collecting information that does not apply.
On the other hand, my knowledge of the author suggests that he included the red dot *somewhere*. That it wasn't in the first half just tells me that he probably didn't use random positioning to choose the red dot's location -- which is more than likely true. So when I continue to believe that 90% of those books do include a red dot, I have the more correct view of the world _regardless of where the author *chooses* to put the dot_.
Basically, my interpretation works for any position the author chooses for the dot. Yours only works if the author is randomly selecting pages.
Yes, and part of my assumption was that the dot is uniformly distributed when it's in the book.
If we have no idea about how the author is putting the dots in the book, no interpretation in the world is going to help. Maybe the author is actively trying to thwart us and always putting it right after where we're going to expect. Maybe there are no red dots at all! But in the real life situation this was based on, I had no reason to believe the red dot was more likely to be at any spot than at any other.
And when I run Bayes Theorem, I get similar non-linear result. newx = (xy) / (xy + z(1-x)) https://plus.google.com/116077417321878358008/posts/Q9bHSEtesxQGiven a 100 page book... x = probability beforehand of dot NOT in book = 0.1 y = probability of first page being blank if the book does not have a red dot = 1 (total certainty) z = probability of first page being blank if the book does have a red dot = 0.99 (because 99 of 100 pages do not have red dot) newx = 0.100908 The probability decreases, as you expected, but not linearly.
Is that if we can't figure out a math problem with the kids' homework, (unlikely with Stephen, I know), I know who I'm calling.
Also, I may have the kids learn calculus or programming or something over the summers at y'all's house, someday... Mwah-ha-ha-ha...
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